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or **muqarnas**

Honeycomb-like Islamic architectural ornamentation formed by the intricate corbeling (*see* corbel) of squinches (*see* Byzantine architecture), brackets, and inverted pyramids in overlapping tiers. It appeared in the early 12th century throughout the Islamic world; a frequent use was to overlay the transitional zone between domes and their supports. It reached its highest development in the 14th–15th century, when it became the usual decoration for door heads, niches, and the bracketing under cornices and minaret galleries. Rich examples are found in the Alhambra and other Moorish works in Spain.

Learn more about stalactite work with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

A honeycomb is a mass of hexagonal wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen.

Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about 8.4 pounds of honey to secrete one pound of wax (Graham 1992), so it makes economic sense to return the wax to the hive after harvesting the honey, commonly called "pulling honey" or "robbing the bees" by beekeepers. The structure of the comb may be left basically intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. Fresh, new comb is sometimes sold and used intact as comb honey, especially if the honey is being spread on bread rather than used in cooking or to sweeten tea.

Broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey. Honeycomb in the "supers" that are not allowed to be used for brood stays light coloured.

Numerous wasps, especially polistinae and vespinae, construct hexagonal prism packed combs made of paper instead of wax; and in some species (like Brachygastra mellifica), honey is stored in the nest, thus technically forming a paper honeycomb. However, the term "honeycomb" is not often used for such structures.

The axes of honeycomb cells are always quasi-horizontal, and the non-angled rows of honeycomb cells are always horizontally (not vertically) aligned. Thus, each cell has two vertical walls, with "floors" and "ceilings" composed of two angled walls. The cells slope slightly upwards, between 9 and 14 degrees, towards the open ends.

There are two possible explanations for the reason that honeycomb is composed of hexagons, rather than any other shape. One, given by Jan Brożek, is that the hexagon tiles the plane with minimal surface area. Thus a hexagonal structure uses the least material to create a lattice of cells within a given volume. Another, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.

The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit three-dimensional and little-noticed. The ends are trihedral (i.e., composed of three planes) pyramidal in shape, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume. (The angle formed by the edges at the pyramidal apex is approximately 109° 28' 16" (= 180° - arccos(1/3)).)

The three-dimensional geometry of a honeycomb cell.

The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells.

Opposing layers of honeycomb cells fit together.

Individual cells do not, of course, show this geometric perfection: in a regular comb, there are deviations of a few percent from the "perfect" hexagonal shape. In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted.

In 1965, László Fejes Tóth discovered that the trihedral pyramidal shape (which is composed of three rhombi) used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or approximately 1 part per 2850) more efficient. This difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, considering that wild comb varies considerably from any mathematical notion of "ideal" geometry.

- Graham, Joe. The Hive and the Honey Bee. Hamilton/IL: Dadant & Sons; 1992; ISBN
- Thompson, D'Arcy Wentworth (1942). On Growth and Form. Dover Publications. ISBN.
- "The Mathematics of the Honeycomb" (June 1985). Science Digest, pp. 74-77.
- "The Royal Institution of Great Britain Christmas Lectures"

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Last updated on Thursday September 11, 2008 at 23:09:59 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday September 11, 2008 at 23:09:59 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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